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OG12-ps-qn-163

by kishokbabu » Fri Jan 06, 2012 12:52 pm
163. This year Henry will save a certain amount of his
income, and he will spend the rest. Next year Henry
will have no income, but for each dollar that he saves
this year, he will have 1 + r dollars available to spend.
In terms of r, what fraction of his income should Henry
save this year so that next year the amount he has
available to spend will be equal to half the amount that
he spends this year?

(A) 1/ (r +2)
(B) 1/ (2r +2)
(C) 1/ (3r +2)
(D) 1/ (r +3)
(E) 1/ (2r +3)

Since i am taking more than 4 min to solve this qn. Is there any time consuming method for this qn?
(I would like to thank GMATGuruNy, thanks for the links that you are providing for my qns)
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by Mike@Magoosh » Fri Jan 06, 2012 2:42 pm
Hi, there! I'm happy to help with this! :-)

I would say the fastest way to think about this is with algebra.

I'm going to use x for the quantity for which the question asks --- x is the proportion of income this year that Henry saves. Of course, 0 < x < 1.

So, Henry has some total income: call this T. This year, he saves xT, and he spends (1 - x)T.

Because he saves xT this year, that gives him (1 + r)xT to spend next year.

This amount that he will spend next year --- (1 + r)xT --- is half of what he spends this year --- (1 - x)T. This gives us the equation:

(1 + r)xT = (1/2)(1 - x)T

Divide both sides by T

(1 + r)x = (1/2)(1 - x)

Multiply by 2 to get rid of the fraction

2x(1 + r) = 1 - x

Distribute on the left

2x + 2xr = 1 - x

Add x to both sides, so all the x's are on one side of the equation.

3x + 2xr = 1

Factor out a factor of x

x(3 + 2r) = 1

Divide by the parentheses:

x = 1/(3 + 2r)

x = 1/(2r + 3) ====> Answer E

Here's another Problem Solving question about ratios, just for additional practice.

https://gmat.magoosh.com/questions/71

Does all this make sense? Please let me know if you have any questions about this.

Mike :-)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
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by karthikpandian19 » Fri Jan 06, 2012 2:43 pm
For easy understanding, let income be I, savings be S

Now, this is a word problem, so interpretation from the problem is very important

What is it asking: "what fraction of income shld he save this yr" = S/I

Now what we have :
1. "This year Henry will save a certain amount of his
income, and he will spend the rest."

I-S = Spending

2. "but for each dollar that he saves
this year, he will have 1 + r dollars available to spend"

S(1+r) = available for nxt yr spending

3. " that next year the amount he has
available to spend will be equal to half the amount that
he spends this year"

S(1+r) = 1/2(I-S)

Now this is final action, Solve it to get S/I

So, here we go

2S(1+r) = I-S

2S+2Sr = I-S

3S+2Sr = I

S(3+2r) = I

So, S/I = 1 / (3+2r)

So the answer is E
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by GMATGuruNY » Fri Jan 06, 2012 2:54 pm
I posted a solution here:

https://www.beatthegmat.com/official-gui ... 61258.html
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by LeoBen » Sun Jan 08, 2012 12:45 pm
variables, i is income, s is saving, o is spending

Matrix for info

-------Year 1-----Year 2

Save----s ----------0
Spend---o --------s(1+r)
Income--i-----------0

wat's asked = s/i


year 1: i = s + o

year 2: i = 0 + s(1+r)

=> given spending year 2 = saving of year 1 (1+r)

give => spending of year 1 = twice of spending available in year 2
=> o = 2s(1+r)

hence, year 1 equation => i = s + s(1+r) [replacing o = 2s(1+r) in year 1 equation]
on solving for s/i u get equation in option E
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